TY - JOUR
T1 - Positive semidefiniteness of estimated covariance matrices in linear models for sample survey data
AU - Haslett, Stephen
N1 - Publisher Copyright:
© 2016 Stephen Haslett, published by De Gruyter Open.
PY - 2016/1
Y1 - 2016/1
N2 - Descriptive analysis of sample survey data estimates means, totals and their variances in a design framework. When analysis is extended to linear models, the standard design-based method for regression parameters includes inverse selection probabilities as weights, ignoring the joint selection probabilities. When joint selection probabilities are included to improve estimation, and the error covariance is not a diagonal matrix, the unbiased sample based estimator of the covariance is the Hadamard product of the population covariance, the elementwise inverse of selection probabilities and joint selection probabilities, and a sample selection matrix of rank equal to the sample size. This Hadamard product is however not always positive definite, which has implications for best linear unbiased estimation. Conditions under which a change in covariance structure leaves BLUEs and/or BLUPs are known. Interestingly, this class of "equivalent" matrices for linear models includes non-positive semi-definite matrices. The paper uses these results to explore how the estimated covariance from the sample can be modified so that it meets necessary conditions to be positive semidefinite, while retaining the property that fitting a linear model to the sampled data yields the same BLUEs and/or BLUPs as when the original Hadamard product is used.
AB - Descriptive analysis of sample survey data estimates means, totals and their variances in a design framework. When analysis is extended to linear models, the standard design-based method for regression parameters includes inverse selection probabilities as weights, ignoring the joint selection probabilities. When joint selection probabilities are included to improve estimation, and the error covariance is not a diagonal matrix, the unbiased sample based estimator of the covariance is the Hadamard product of the population covariance, the elementwise inverse of selection probabilities and joint selection probabilities, and a sample selection matrix of rank equal to the sample size. This Hadamard product is however not always positive definite, which has implications for best linear unbiased estimation. Conditions under which a change in covariance structure leaves BLUEs and/or BLUPs are known. Interestingly, this class of "equivalent" matrices for linear models includes non-positive semi-definite matrices. The paper uses these results to explore how the estimated covariance from the sample can be modified so that it meets necessary conditions to be positive semidefinite, while retaining the property that fitting a linear model to the sampled data yields the same BLUEs and/or BLUPs as when the original Hadamard product is used.
UR - http://www.scopus.com/inward/record.url?scp=85011961761&partnerID=8YFLogxK
U2 - 10.1515/spma-2016-0020
DO - 10.1515/spma-2016-0020
M3 - Article
SN - 2300-7451
VL - 4
SP - 218
EP - 224
JO - Special Matrices
JF - Special Matrices
IS - 1
ER -